Séance Séminaire

Séminaire ACSIOM

Modelling the dynamics of ice sheets using an adaptive multigrid method

For more than one decade, Antarctic ice sheets have been retreating dramatically, and are expected to shrink even more quickly in the future. Recent studies have identified the migration of the grounding lines separating the grounded part of ice sheets and the floating part of ice shelves as being a key process controlling marine ice sheet stability. It is also well-known that the grounding line is very sensitive with respect to small climatic perturbations. Over an upward-sloping bedrock, this sensitivity induces fast and irreversible retreats of the grounding line and, then, of the ice sheet. In this talk, we consider a model for the time evolution of ice sheets and ice shelves. First, the slow deformation of ice – which dominates on the grounded part – is described by the the Shallow Ice Approximation (SIA). Second, the fast basal sliding – which dominates on the floating part – is described by the Shallow Shelf Approximation (SSA). At each time step, we have to solve one scalar generalized p-Laplace problem with obstacle and p > 2 (SIA) [2] and one vectorial p-Laplace problem with 1 < p < 2 (SSA) [3]. Both problems can be advantageously rewritten by minimising suitable, convex non-smooth energies. By exploiting such formulations, we implement a truncated non-smooth Newton multigrid method [1]. Our approach allows a wide choice of unstructured meshes to be used and can be easily combined to mesh adaptation techniques. In practice, one needs to refine the mesh in the neighbourhood of the grounding line (transition zone) to capture the high gradients due to the sharp changes in the dynamical regime of ice. To optimize automatically the number and the position of mesh nodes, we implement a re-meshing procedure based on a hierarchical error estimate for the SSA equation. As an illustration, we present numerical results based on the exercises of the Marine Ice Sheet Model Inter-comparison Project (MISMIP). References [1] C. Gräser and R. Kornhuber. Multigrid methods for obstacle problems. J. of Comp. Math., 27(1):1–44, 2009. [2] G. Jouvet and E. Bueler. Steady, shallow ice sheets as obstacle problems: well-posedness and finite element approximation. SIAM J. Appl. Math. 72(4), 1292–1314, 2012. [3] C. Schoof. A variational approach to ice stream flow. J. Fluid Mech., 556, 2006.